Discontinuous Galerkin approximation of flows in fractured porous media on polytopic grids

We present a numerical approximation of Darcy's flow through a fractured porous medium which employs discontinuous Galerkin methods on polytopic grids. For simplicity, we analyze the case of a single fracture represented by a (d 1)-dimensional interface between two d-dimensional subdomains, d = 2, 3. We propose a discontinuous Galerkin finite element approximation for the flow in the porous matrix which is coupled with a conforming finite element scheme for the flow in the fracture. Suitable (physically consistent) coupling conditions complete the model. We theoretically analyze the resulting formulation, prove its well-posedness, and derive optimal a priori error estimates in a suitable (mesh-dependent) energy norm. Two-dimensional numerical experiments are reported to assess the theoretical results.